Equations of State of Ca-Silicates and Phase Diagram of the CaSiO₃ System under Upper Mantle Conditions

Abstract

The equations of state of different phases in the CaSiO₃ system (wollastonite, pseudowollastonite, breyite (walstromite), larnite (Ca₂SiO₄), titanite-structured CaSi₂O₅ and CaSiO₃-perovskite) are constructed using a thermodynamic model based on the Helmholtz free energy. We used known experimental measurements of heat capacity, enthalpy, and thermal expansion at zero pressure and high temperatures, and volume measurements at different pressures and temperatures for calculation of parameters of equations of state of studied Ca-silicates. The used thermodynamic model has allowed us to calculate a full set of thermodynamic properties (entropy, heat capacity, bulk moduli, thermal expansion, Gibbs energy, etc.) of Ca-silicates in a wide range of pressures and temperatures. The phase diagram of the CaSiO₃ system is constructed at pressures up to 20 GPa and temperatures up to 2000 K and clarifies the phase boundaries of Ca-silicates under upper mantle conditions. The calculated wollastonite–breyite equilibrium line corresponds to equation P(GPa) = −4.7 × T(K) + 3.14. The calculated density and adiabatic bulk modulus of CaSiO₃-perovskite is compared with the PREM model. The calcium content in the perovskite composition will increase the density of mineral and it good agree with the density according to the PREM model at the lower mantle region.

1. Introduction

The thermodynamic description and phase equilibria in the CaSiO₃ system are important for modern mineralogical research because Ca-silicates are common components of the Earth’s crust and mantle. The low-pressure CaSiO₃ polymorphs (wollastonite and pseudowollastonite) are rock-forming minerals in the Earth’s crust and constitute cement substances [1,2], while the high-pressure Ca-silicate phases (breyite (walstromite), larnite, titanite-structured CaSi₂O₅ and CaSiO₃-perovskite) are stable in the mantle P-T conditions and detected as solid-phase inclusions in natural diamonds [3,4,5,6,7]. The presence of CaSiO₃ phase in diamonds indicates the possibility of transporting the substance from the lower mantle to the surface in an unchanged form and provides insights into the nature of diamond formation. Thus, the present study will expand the possibility for thermodynamic modeling of physicochemical processes in the Earth’s deep mantle.

It is well known that calcium metasilicate (CaSiO₃) has a number of structural modifications at relevant pressures and temperatures. Wollastonite structure is stable under ambient conditions and transforms to pseudowollastonite phase at high temperatures above ~1400 K [8]. The transition of wollastonite to walstromite-structured CaSiO₃ is detected at 3 GPa and 1173 K [9]. The name breyite for this phase is approved by IMA in 2018 [10], therefore it is used in the text below. The wollastonite–breyite equilibrium was studied by different authors [1,11,12]. The subsequent decomposition of breyite to association of larnite (Ca₂SiO₄) and titanite-structured CaSi₂O₅ was determined at higher pressures by Kanazaki’s study [13]. The CaSiO₃-perovskite is formed at pressure ~14 GPa from (larnite + CaSi₂O₅) assemblage [14] and retains its structure until lower mantle conditions. The phase diagram of the CaSiO₃ is well studied at low pressures up to 3–4 GPa and high temperatures by experimental methods in [11,15]. In the present calculation, we also used the phase diagram of the CaSiO₃ system, which was proposed in [1,11].

The thermodynamics of Ca-silicates can be calculated from their equations of state (EoS). The equations of state of wollastonite, pseudowollastonite, breyite, larnite (Ca₂SiO₄), titanite-structured CaSi₂O₅ and CaSiO₃-perovskite can be constructed based on the thermodynamic model, which was successfully used in our previous studies for the MgSiO₃–MgO system [16,17]. We use a thermodynamic model based on the Helmholtz free energy, which allows us to calculate a full set of thermodynamic properties (entropy, heat capacity, bulk moduli, thermal expansion, Gibbs energy, etc.) at given P-T or V-T parameters. The proposed thermodynamic model is based on optimization of different experimental measurements (dilatometric, X-ray diffraction, ultrasonic interferometry, etc.) and theoretical data (calculations and reference) by analogy with our studies for metals [18,19,20] and compounds [21,22]. The proposed equations of state of studied Ca-silicates are constructed with a small number of fitting parameters. There are also other different forms of EoSes, which contain a small number of parameters and reliably describe the properties of metals and mixtures [23,24,25,26], but in this study, we will use our model that previously showed its versatility.

2. Thermodynamic Model

The equations of state of wollastonite, pseudowollastonite, breyite, larnite (Ca₂SiO₄), titanite-structured CaSi₂O₅ and CaSiO₃-perovskite are constructed using the thermodynamic model based on the Helmholtz free energy. According to the proposed thermodynamic model, the Helmholtz free energy can be represented in general form as the sum [27]:

$$F\left(V,T\right)={\mathcal{U}}_0+F_{T_0}\left(V\right)+F_{th}\left(V,T\right)+F_{anh}\left(V,T\right),$$

(1)

where U₀ is the reference energy, F_T0 is the potential part of the Helmholtz free energy at the reference isotherm T₀ = 298.15 K, which depends only on volume; F_th is the thermal part of the Helmholtz free energy, which depends on volume and temperature; F_anh is an additional contribution to the Helmholtz free energy, which connects with intrinsic anharmonicity, as a function of temperature and volume.

The potential part of the Helmholtz free energy from Equation (1) is determined by the integration of the equation of pressure at the reference isotherm. Many different equations are known to express the equation of pressure at the reference isotherm, each of which has its advantages and limitations (review in [17]). We use the Kunc equation [28], which is flexible in calculations because it contains parameter k:

$$P_{T_0}\left(V\right)=3K_0{X}^{-k}\left(1-X\right)\exp \left[\eta \left(1-X\right)\right],$$

(2)

where X = (V/V₀)^1/3, η = 1.5K′ − k + 0.5, K₀ is the isothermal bulk modulus under ambient conditions (P₀ = 1 bar = 10⁻⁴ GPa, T₀ = 298.15 K), K′ = ∂K₀/∂P₀ and k is an additional parameter.

When k = 2, the Kunc equation corresponds to the Vinet equation [29], which is often used in calculation of equations of state of metals in solid state physics. The values of k = 5 and k > 5 transform the Kunc equation into one of the forms of the Holzapfel equation (HO₂) [30] and into the Birch–Murnaghan equation [31], respectively. We use k = 5 for all Ca-silicate phases in the present study.

Differentiating Equation (2) with respect to volume at constant temperature, we determine the isothermal bulk modulus at the reference isotherm and calculate of its derivation by pressure:

$$K_{T0}\left(V\right)=-V{\left(\frac{\partial P_{T_0}}{\partial V}\right)}_T=K_0{X}^{-k}\exp \left[\eta \left(1-X\right)\right]\left[X+\left(1-X\right)\left(\eta X+k\right)\right],$$

(3)

$$K^{\prime }\left(V\right)=\left(\frac{\partial K_{T0}}{\partial P_{T_0}}\right)=\frac{1}{3}\left[k+\eta X+\frac{kX+2X^2\eta -X\left(1+\eta \right)}{X+\left(1-X\right)\left(\eta X+k\right)}\right].$$

(4)

The thermal part of the Helmholtz free energy from Equation (1) can be calculated by Debye, Einstein, Bose–Einstein models, or their combinations [32,33]. A more simple approach is to use the Einstein model with two characteristic temperatures, which makes it possible to good approximate of the heat capacity in the temperature range from ~100 K to the melting point of the substance:

$$F_{th}\left(V,T\right)={\displaystyle \sum }_{i=1,2}{m}_{i}RT\ln \left(1-\exp \frac{-{\mathsf{\Theta }}_i}{T}\right),$$

(5)

where m_i is proportionality factor, which is calculated from the total number of atoms in the compound (i = 1, 2), m₁ + m₂ = 3n, n is the number of atoms in a chemical formula of the compound, Θ_i is the Einstein characteristic temperature, which depends only on volume, and R is the gas constant (R = 8.31451 Jmol⁻¹K⁻¹).

The volume dependence of characteristic temperature in Equation (5) is determined by Al’tshuler equation [34]:

$${\mathsf{\Theta }}_i={\mathsf{\Theta }}_{0i}{x}^{-{\gamma }_{\infty }}\exp \left[\frac{{\gamma }_0-{\gamma }_{\infty }}{\beta }\left(1-x^{\beta }\right)\right],$$

(6)

where Θ_0i is the characteristic temperature under ambient conditions (i = 1, 2), x = V/V₀, γ₀ is the Grüneisen parameter at ambient conditions, γ_∞ is the Grüneisen parameter at infinite compression, when x → 0, and β is an additional fitting parameter.

The advantage of Equation (6) is that it can be explicitly differentiated in the simple form to determine the Grüneisen parameter:

$$\gamma =-{\left(\frac{\partial \ln \mathsf{\Theta }}{\partial \ln V}\right)}_T={\gamma }_{\infty }+\left({\gamma }_0-{\gamma }_{\infty }\right)x^{\beta },$$

(7)

where Θ is the characteristic temperature in general case.

We assume that the Grüneisen parameter will be the same for both characteristic temperatures (Θ₁ and Θ₂) in Equation (6). In addition, if γ_∞ = 0, then Equation (7) transforms to the classical form γ = γ₀x^β.

Differentiating Equation (5) with respect to temperature at constant volume, we determine entropy and then calculate internal energy:

$$S=-{\left(\frac{\partial F_{th}}{\partial T}\right)}_V={\displaystyle \sum }_{i=1,2}3nR\left[-\ln \left(1-\exp \frac{-{\mathsf{\Theta }}_i}{T}\right)+\frac{{\mathsf{\Theta }}_i/T}{\exp \left({\mathsf{\Theta }}_i/T\right)-1}\right],$$

(8)

$$E_{th}=F_{th}+TS={\displaystyle \sum }_{i=1,2}3nR\left[\frac{{\mathsf{\Theta }}_i}{\exp \left({\mathsf{\Theta }}_i/T\right)-1}\right].$$

(9)

Differentiating Equation (5) with respect to volume at constant temperature, we obtain the thermal part of pressure:

$$P_{th}=-{\left(\frac{\partial F_{th}}{\partial V}\right)}_T={\displaystyle \sum }_{i=1,2}3nR\frac{\gamma }{V}\left[\frac{{\mathsf{\Theta }}_i}{\exp \left({\mathsf{\Theta }}_i/T\right)-1}\right]=\frac{\gamma }{V}{E}_{th}.$$

(10)

Differentiating Equation (9) with respect to temperature at constant volume and Equation (10) with respect to volume at constant temperature, we obtain the isochoric heat capacity and thermal part of isothermal bulk modulus:

$$C_V={\left(\frac{\partial E_{th}}{\partial T}\right)}_V={\displaystyle \sum }_{i=1,2}3nR{\left(\frac{{\mathsf{\Theta }}_i}{T}\right)}^2\frac{\exp \left({\mathsf{\Theta }}_i/T\right)}{{\left(\exp \left({\mathsf{\Theta }}_i/T\right)-1\right)}^2},$$

(11)

$$K_{Tth}=-V{\left(\frac{\partial P_{th}}{\partial V}\right)}_T=P_{th}\left(1+\gamma -q\right)-{\gamma }^2{C}_V\frac{T}{V},$$

(12)

where q is additional parameter, which is calculated from Equation (7) according to [34].

Differentiating Equation (10) with respect to temperature at constant volume, we determine the slope:

$${\left(\partial P_{th}/\partial T\right)}_V=\frac{\gamma }{V}{C}_V.$$

(13)

The additional contribution of the intrinsic anharmonicity to the Helmholtz free energy in Equation (1) can be expressed by classical equation from [27]:

$$F_{anh}\left(V,T\right)=-\frac{3}{2}nRa{T}^2=-\frac{3}{2}nR{a}_0{x}^m{T}^2.$$

(14)

The contribution of the intrinsic anharmonicity to entropy, internal energy, thermal part of pressure, isochoric heat capacity, thermal part of isothermal bulk modulus and slope can be obtained by analogy with Equations (8)–(13), respectively:

$$\begin{array}{c}{S}_{anh}= 3nRaT,\\ E_{anh}=\frac{3}{2}nRa{T}^2,\\ C_{V_{anh}}= 3nRaT,\\ P_{anh}=\frac{3}{2}nRa\frac{m}{V}{T}^2= \frac{m}{V}{E}_{anh},\\ K_{T_{anh}}= P_{anh}\left(1-m\right),\\ {\left(\partial P_{anh}/\partial T\right)}_V=\frac{m}{V}{C}_{V_{anh}},\end{array}$$

(15)

where a = a₀x^m, a₀ is the intrinsic anharmonicity parameter and m is the anharmonic analogue of the Grüneisen parameter.

The Equation (10) for the thermal part of pressure and equation for anharmonicity part of pressure (P_anh in Equation (15)), as can be seen, are very similar. The Grüneisen parameter in Equation (10) is the coefficient of proportionality between pressure and internal energy, just like the parameter m in Equation (15). Therefore, we use this name for the parameter m. The parameters a₀ and m are fitting parameters in the general cause, but we use the fixed value for m (m = 1) in the present study.

The full equations of pressure and isothermal bulk modulus are calculated as the sum of potential, thermal, and anharmonicity components: P = P_T0(V) + P_th(V,T) + P_anh(V,T) and K_T = K_T0(V) + K_Tth(V,T) + K_Tanh(V,T). Further, it is easy to calculate the coefficient of thermal expansion α = (∂P_th/∂T)_V / K_T, the heat capacity at constant pressure C_P = C_V + α²TVK_T, the adiabatic bulk modulus K_S = K_T + VT(αK_T)²/C_V, and the thermodynamic Grüneisen parameter γ_th = αVK_T/C_V = αVK_S/C_P. The enthalpy and Gibbs energy are determined from the linear relations H = E_th + PV − P₀V₀ and G = F(V,T) + PV − P₀V₀, respectively. The parameter U₀ in Equation (1) define the Gibbs energy under ambient conditions, which also can be found from the simple relation: U₀ = ∆G_f,₂₉₈ = ΔH_f,₂₉₈ − S₂₉₈T₀, where ΔH_f,₂₉₈ and S₂₉₈ are enthalpy and entropy at ambient conditions, respectively.

Thus, Equations (1)–(15) contain a group of fixed parameters (V₀, k, m_i) and a group of fitting parameters (K₀, K’, Θ_0i, γ₀, γ_∞, β, a₀, m) derived by least squares method in Excel worksheet. Using the proposed approach, it is easy to calculate a full set of thermodynamic functions of Ca-silicates that depend on temperature, pressure, and volume.

3. Results

The equations of state of studied Ca-silicates were constructed using thermodynamic model described above.

Table 1. Parameters of equations of state of Ca-silicates at ambient conditions.

Parameters	Wol CaSiO3	PsWol CaSiO3	Brt CaSiO3	β-Lrn Ca2SiO4	Ca-Tit CaSi2O5	Ca-Pv CaSiO3
𝓤0, kJmol−1	−1658.007	−1655.531	−1651.400	−2344.025	−2524.000	−1567.400
V0, cm3mol−1	39.9	40.3	37.78	51.88	48.19	27.403
K0, GPa	81.5	86	78.6	105	178	239
K’	3.28	3.8	4	6.27	4	4.13
k	5	5	5	5	5	5
Θ01, K	966	1039	952	838	975	935
m1	7.5	7.5	7.5	10.5	12	7.5
Θ02, K	270	244	280	257	370	388
m2	7.5	7.5	7.5	10.5	12	7.5
γ0	1	1	0.745	1.263	1.3	1.913
γ∞	0	0	0	0	0	0
β	1	1	1	1.041	1	0.486
a0, 10−6 K−1			20.8
m			1

shows parameters of equations of state for wollastonite (Wol), pseudowollastonite (PsWol), breyite (Brt), larnite (β-Lrn), titanite-structured CaSi₂O₅ (Ca-Tit) and CaSiO₃-perovskite (Ca-Pv) obtained by simultaneous optimization of known thermochemical measurements of the heat capacity, enthalpy, thermal expansion, and numerous P-V-T data including modern X-ray diffraction measurements for the Ca-silicate minerals. The optimization procedure is described in detail in our previous works [16,20,33]. In the present study, we also offer the open working MS Excel spreadsheets for each of the studied Ca-silicates (Files S1–S6 in Supplementary Material). Detailed instructions and descriptions of worksheets are provided in [17,19]. Therefore, below we will focus on the features of calculating of equation of state for each of the phases.

Wollastonite (Wol). A review of the heat capacity measurements of wollastonite and other CaSiO₃ phases can be found in [35]. The most reliable measurements we found of molar volume of wollastonite as a function of temperature are published in [36]. The only known measurement of the volume of wollastonite as a function of pressure in [37] shows the following values of K₀ = 54.7 GPa and ∂K/∂P = 23.5. However, at these values, our equation of state of wollastonite becomes unstable at temperatures above 700 K (this is easily verified by substituting the parameters to the present Excel file for wollastonite). Therefore, we use measurements of volume, depending on pressure for pseudowollastonite from [38]. The simultaneous optimization of known experimental data for wollastonite at γ₀ = 1, β = 1 and γ_∞ = 0 determine the obtained parameters of its equation of state (Table 1). Figure 1a shows a comparison of the calculated isobaric (C_P) and isochoric (C_V) heat capacity of wollastonite with measurements from [39,40,41,42]; Figure 1c shows calculated molar volume at reference pressure (P₀ = 1 bar = 10⁻⁴ GPa) with Swamy’s data [36]; and Figure 1e shows calculated compressibility as a function of pressure with measurements from [37]. It can be seen that calculated data are in good agreement with different reference and experimental data.

The calculated thermodynamic functions of wollastonite at different temperatures and pressures (0.0001 and 3 GPa) are presented in

Table 2. Thermodynamic functions of wollastonite at different pressures and temperatures.

P, GPa	T, K	V, cm3 mol−1	α × 106, K−1	S, Jmol−1 K−1	CP, Jmol−1 K−1	CV, Jmol−1 K−1	KT, GPa	KS, GPa	ΔG, kJmol−1
0.0001	298.15	39.900	26.458	81.358	86.715	86.037	81.50	82.14	−1658.007
0.0001	500	40.147	33.714	132.366	108.976	107.159	79.66	81.01	−1679.826
0.0001	1000	40.907	40.288	214.108	124.900	119.946	74.62	77.70	−1768.341
0.0001	1398	41.596	43.519	256.827	130.068	122.310	70.44	74.91	−1862.474
0.0001	2000	42.763	48.446	304.482	136.431	123.596	63.94	70.58	−2032.169
3	298.15	38.536	23.175	78.447	84.890	84.327	91.20	91.81	−1540.406
3	500	38.746	29.732	128.623	107.621	106.089	89.43	90.72	−1561.542
3	1000	39.393	35.446	209.562	123.771	119.586	84.55	87.51	−1647.956
3	1500	40.130	38.619	260.949	129.599	122.464	79.48	84.11	−1766.352
3	2000	40.944	41.668	298.850	134.050	123.490	74.28	80.63	−1906.702

. The last column shows the Gibbs energy at given P-T parameters. The parameter U₀ is calculated from the enthalpy at ambient conditions ΔH_f,298 = −1633.750 kJmol⁻¹ from [43] and entropy S₂₉₈ = 81.358 Jmol⁻¹ K⁻¹ at ambient conditions from Table 2, then U₀ = ∆G_f,298 = −1,633,750 − 81.358 × 298.15 = −1,658,007 Jmol⁻¹. The same scheme was used for calculation of the Gibbs energy for Mg-silicates in the Mg₂SiO₄ system [16]. We have shown calculation thermodynamic functions for wollastonite in the Table 2. Further tabulated values of calculated thermodynamic functions for other Ca-silicates are presented in the Supplementary Material (Tables S1–S5).

Pseudowollastonite (PsWol). As follows from the review in [2], wollastonite at ambient conditions has a triclinic structure (space group P1). The crystal structure of a synthetic two-layer polytype of pseudowollastonite was studied using single-crystal X-ray diffraction data at pressures up to 10 GPa by Yang and Prewitt [38]. It is determined that such a polytype has monoclinic structure (space group C2/c) and molar volume at ambient condition (V₀ = 796.9 A³ = 39.99 cm³mol⁻¹) coincides with the volume of wollastonite (Table 2). The molar volume of pseudowollastonite was studied at temperature range 307–1793 K by Richet et al. [44]. There are two-, four-, and six-layer polytypes of CaSiO₃-pseudowollastonite in this temperature range. The parameters of K₀ and K’ for our calculation were extracted from [38]; parameters of Θ₀₁ and Θ₀₂ are calculated from the heat capacity and relative enthalpy measurements from [42,45,46,47]. The parameter U₀ is calculated from the condition of equality of the Gibbs energy of wollastonite and pseudowollastonite at T = 1398 K and P = 0.0001 GPa. The difference of calculated entropy at 1398 K for wollastonite–pseudowollastonite transition is ∆S₁₃₉₈ = 1.586 Jmol⁻¹K⁻¹ (see Table 2 and Table S1), while in Richet’s study, it is ∆S₁₃₉₈ = 4.2 ± 2 Jmol⁻¹K⁻¹ [42]. However, taking into account ∆V₁₃₉₈ = 0.3036 cm³mol⁻¹, we obtain the slope ∂P/∂T = ∆S/∆V = 52.2 × 10⁻⁴ GPa/K, which is in good agreement with data from [11,48].

The calculated thermodynamic functions of pseudowollastonite are compared with different reference and experimental data [38,44] by analogy with wollastonite on Figure 1b,d,f. The tabulated values of calculated thermodynamic functions of pseudowollastonite as a function of temperature at different pressures (0.0001 and 3 GPa) is presented in the Supplementary Material (Table S1).

Breyite (Brt). Regarding of breyite, in the literature, we can find the different names of its structural isomorphs (Ca-walstromite, wollastonite-II, walstromite-like CaSiO₃(III)). The crystal structures of breyite (or walstromite), CaSiO₃-walstromite, and wollastonite-II are very similar. This issue is discussed in detail by Joswig et al. [49]. Breyite probably crystallizes in the triclinic structure (space group P1) and it is stable in the pressure range from ~3 GPa and above, since the remaining pressure of investigated inclusions of breyite in natural diamonds [3,4] was also estimated to be 3 GPa. The direct P-V-T measurements of breyite are carried out in [5,50]. The parameters of V₀, K₀ and K’ are taken from Anzolini et al. [5]. The parameter U₀ is calculated from the phase transition of wollastonite–breyite. The calculated volume of breyite as a function of temperature and pressure is compared with experimental data on Figure 2a,b. The tabulated values of calculated thermodynamic functions of breyite depending on temperature at different pressures (0.0001, 5 and 10 GPa) is presented in the Supplementary Material (Table S2).

Larnite (β-Lrn). As temperature increases, the composition of Ca₂SiO₄ can form a series of polymorphs (γ-, β-, α′_L-, α’_H- and α). The review of crystal structures of Ca₂SiO₄ is presented for example in [51,52]. The β-larnite is formed as metastable monoclinic phase (space group P2₁/n11) in the stability field of γ-Ca₂SiO₄ by its cooling. We considered here only the beta modification, which is typical for natural larnite [53]. Larnite (β phase) has a wide stability field at high pressures and temperatures; therefore, this phase is used in the present study. The parameter V₀ is taken from Remy et al. [51], and other parameters (K₀, K’, Θ_0i, γ₀, γ_∞ and β) are calculated from optimization of experimental measurements of heat capacity and P-V-T data from [51,54,55,56,57]. The parameter U₀ is calculated from the enthalpy at ambient conditions ΔH_f,298 = −2307.04 kJmol⁻¹ from [43] and entropy at ambient conditions S₂₉₈ = 124.047 Jmol⁻¹K⁻¹ from Table S3 (Supplementary Material). The other tabulated values of calculated thermodynamic functions of larnite depending on temperature at different pressures (0.0001, 10 and 15 GPa) is presented also in Table S3. The calculated volume of larnite as a function of pressure and temperature is compared with experimental data on Figure 3a,b.

Titanite-structured CaSi₂O₅ (Ca-Tit). The titanite-structured CaSi₂O₅ phase is identified by in situ XRD analyses in natural diamonds from Guinea and Brazil [3,4,5], but thermochemical data for this phase are not available. Therefore, the present equation of state of titanite-structured CaSi₂O₅ (space group C2/c) is constructed on the values of parameters of V₀, K₀ and K’ from [58], and other parameters (Θ_0i, γ₀, γ_∞ and β) are calculated from optimization of experimental data from [2,58]. The parameter U₀ for Ca-titanite is calculated from values of enthalpy and entropy at ambient conditions with analogy for wollastonite and larnite calculations. The tabulated values of calculated thermodynamic functions of titanite-structure CaSi₂O₅ as a function of temperature at different pressures (0.0001, 10 and 15 GPa) is presented in the Supplementary Material (Table S4).

Perovskite (Ca-Pv). CaSiO₃-perovskite is considered as the one of the main Ca-bearing phase in the Earth’s lower mantle [59]. Therefore, findings of Ca-perovskite in natural diamond [6] may explain the mechanisms of diamond formation in general. The question of nature of “super-deep” diamond is discussed widely; however, the aim of this study is to reliably calculate the thermodynamic functions of CaSiO₃-perovskite in a wide range of pressures and temperatures. CaSiO₃-perovskite is stabilized in the cubic structure (space group Pm3m) at temperatures above 1000 K and pressures above 11–16 GPa, according to [1]. At low temperatures up to 500 K and high pressures up to 156 GPa, the CaSiO₃-perovskite has a distorted cubic structure, which is identified in [60,61] as tetragonal modification (space group I4/mcm). The direct measurements of heat capacity and bulk modulus as a function of temperature at ambient pressure for cubic Ca-perovskite are not available [62]; therefore, all parameters (K₀, K’, Θ_0i, γ₀, γ_∞ and β) for proposed equation of state were calculated by authors. However, numerous experimental measurements of molar volume of Ca-perovskite at different pressures and temperatures are known (

Table 3. Direct experimental measurements of molar volume of CaSiO₃-perovskite.

Pressure Range, GPa	Temperature Range, K	Method	Reference
0–134	300	XRD	[68]
16.8–108	300	XRD	[69]
28–62	300	XRD	[61]
0.59–13	301–1600	XRD	[70]
18–96	1238–2419	XRD	[71]
51–130	700–2300	XRD	[63]
21.2–156	1200–2600	XRD	[60]
12–23	700–1700	XRD	[64]

). The proposed equation of state of CaSiO₃-perovskite is constructed on the modern P-V-T measurement from [60,61,63] and simultaneous sound velocities and density measurements at pressures up to 23 GPa and temperatures up to 1700 K from [64]. The pressure measurements in [63] were recalculated based on the equation of state of Pt from [65]; the pressure from [60] was calculated using a self-consistent pressure scale of Pt from [66]; and pressure from [64] was monitored during the experiment by the unit-cell volume of NaCl [67].

The differences in measurements of volume in Table 3 are significant, so the proposed equation of state of Ca-perovskite here is important. Figure 4a,b shows the deviations of the experimental measured pressure in [60,63] and calculated pressure from the proposed equation of state. It can be seen that the most differences are observed at pressure over 100 GPa. Figure 5a,b shows a difference between calculated pressure and adiabatic bulk modulus and experimental data from [64]. The tabulated values of calculated thermodynamic functions of cubic CaSiO₃-perovskite as a function of temperature at different pressures (0.0001, 10, 15 and 100 GPa) are presented in the Supplementary Material (Table S5).

4. Discussion

The proposed equations of state of wollastonite, pseudowollastonite, breyite (walstromite), larnite, titanite-structured CaSi₂O₅ and CaSiO₃-perovskite allow us to calculate the phase diagram of the CaSiO₃ system (Figure 6). The wollastonite–pseudowollastonite equilibrium corresponds to the data from [11,48], which were used for calibration of the Gibbs energy of pseudowollastonite. At a pressure of about 3 GPa and temperature of about 1100 K wollastonite transforms to breyite (walstromite). The Gibbs energy of breyite is calibrated based on measurements from [11,15]. The calculated wollastonite–breyite equilibrium line coincides with the line P(GPa) = −4.7 × T(K) + 3.14, that was recommended by Essene [11] (Figure 6). At higher pressures, breyite transforms to the two-phase assembly of larnite (Ca₂SiO₄) and CaSi₂O₅ [1]. Based on this phase transition, the sum of the Gibbs energy of both phases is calculated. The phase boundary of breyite–(larnite + CaSi₂O₅) is calculated from equation 3Brt = β-Lrn + CaSi₂O₅ [12,57], using equilibrium of the Gibbs energy of breyite, larnite and Ca-titanite. After that, we can estimate the Gibbs energy of Ca-perovskite and calculate the line of phase transition from equation β-Lrn + CaSi₂O₅ = 3Ca-Pv [12,57]. It should be noted that the position of the calculated equilibrium lines (breyite–(larnite + CaSi₂O₅) and (larnite + CaSi₂O₅)–Ca-perovskite) depends entirely on the choice of the primary experimental data, which have a very significant scatter (see Figure 6). We performed a calibration of the Gibbs energy of Ca-perovskite based on the thermochemical data from [12], because we consider these data to be the most reliable. The stability field of high-pressure Ca-silicates were studied by [71,72] and significant deviations were determined. Thus, the obtained phase diagram of the CaSiO₃ system is based on modern P-V-T measurements for Ca-silicates and clarifies the phase boundaries under upper mantle conditions at the depth up to ~600 km.

There are many first-principles calculations based on density functional theory (DFT) for CaSiO₃-perovskite [73,74,75,76], however some of these were made at zero temperature [75,76]. The equation of state of CaSiO₃-perovskite based on ab initio molecular dynamic (AIMD) simulations was constructed in [74]. The authors believe that tetragonal structure of Ca-perovskite is stable at temperature range 1000–4000 K and pressure range 15–130 GPa. However, this assumption does not agree well with the melting line calculations in the CaSiO₃ system (see Figure 7). First-principle molecular dynamics calculations to investigate the structure of Ca-perovskite at high temperatures and pressures were made in [77]. The cubic structure of CaSiO₃-perovskite was found to be stable at temperatures higher 1000 K and at all pressures according to [77]. The phase boundaries in Ca-perovskite based on LDA and GGA approximations were calculated by Stixrude et al. [73]. The location of the phase transition from tetragonal to cubic structure in Ca-perovskite is shown in Figure 7. The gray area indicates the stability field of the tetragonal structure of Ca-perovskite according to [73,77].

The melting line in the CaSiO₃ system was studied experimentally at pressure up to 58 GPa in [1,78,79] and based on first-principles calculations in [80,81]. Figure 7 shows the comparison of the experimental data and different calculations. As can be seen, the experimental data from [79] are in good agreement with calculations [80,81] at the temperature range 2750–3500 K and at the pressure range 15–30 GPa. The melting line from [78] is lower than other estimates and difference increases at high pressures. The melting of CaSiO₃-perovskite is predicted in the range of very high temperatures and pressures up to 6400 K and 300 GPa, respectively, according to the recent study [81]. The melting temperature of CaSiO₃-perovskite significantly increases with pressure and is higher than that of MgSiO₃-perovskite.

The proposed equation of state of Ca-perovskite is of great interest for deep mineralogy. The thermodynamic parameters of the Earth’s lower mantle can be estimated using the equation of state of Ca-perovskite.

Table 4. Density and adiabatic bulk modulus of cubic CaSiO₃-perovskite at temperature, pressure, and depth of the lower mantle from the Preliminary Reference Earth Model (PREM) model.

Depth, km	P, GPa	T, K	Density, gcm−3	Ks, GPa
2891	135.75	3739	5.483	686.56
2771	128.71	2838	5.499	673.01
2741	126.97	2740	5.492	668.35
2571	117.35	2668	5.412	637.46
2371	106.39	2596	5.317	601.82
2171	95.76	2525	5.220	566.86
1971	85.43	2452	5.122	532.51
1771	75.36	2379	5.022	498.60
1571	65.52	2302	4.920	465.04
1371	55.9	2227	4.814	431.71
1171	46.46	2144	4.705	398.53
971	37.29	2060	4.592	365.70
771	28.29	1974	4.474	332.80
670	23.83	1931	4.412	316.19

shows the parameters of the Earth’s lower mantle according to the PREM model (Preliminary Reference Earth Model) [82], where temperature is calculated from [83]. We calculated the density and adiabatic bulk modulus of CaSiO₃-perovskite at these P-T parameters (from Table 4) and compared it with the PREM model and with the density and adiabatic bulk modulus of MgSiO₃-perovskite from our previous study [16] (Figure 8). It can be seen that the density of Ca-perovskite is higher than the density of Mg-perovskite by about 0.1–0.15 gcm⁻³ and almost coincides with the density according to the PREM model (Figure 8a). The adiabatic bulk moduli of Ca-perovskite and Mg-perovskite differs slightly. However, with increasing pressure, adiabatic bulk modulus of CaSiO₃-perovskite exceeds the PREM model by 40 GPa at pressures of 120–130 GPa (Figure 8b). Moreover, we added the first-principle calculated data for the density and adiabatic bulk modulus of Ca-perovskite according to Li et al. [74]. The calculated density from [74] and calculated density from our equation of state for CaSiO₃-perovskite are very similar, especially in the high pressure range above 80 GPa (Figure 8a). The calculated values of the adiabatic bulk modulus are in less agreement (Figure 8b).

Thus, the obtained results of the present study can be used to interpret of the phase and seismic boundaries in the Earth’s upper mantle. Since calcium is one of the main components of the Earth’s mantle, it has its influence on the location of the observed seismic boundaries. At a depth of about ~350 km, CaSiO₃ phase is contained as a component of clinopyroxene minerals, whereas below 350 km, it is concentrated as a component of complex garnet solid solutions (majorite garnet) [9,59]. The phase of CaSiO₃-perovskite is stabilized at increasing pressure and is considered one of the main minerals of the lower mantle, comprising up to 7 wt.% of peridotitic mantle, according to the modern geophysical data [84]. There is the view that the transition of the CaSiO₃ component in the garnet solid solution to form denser of Ca-perovskite is associated with seismic discontinuity at a depth of 520 km [59]. The phase transition of wadsleyite–ringwoodite in the Mg₂SiO₄ system also occurs at this depth at pressure 17–18 GPa, but it is characterized by a slight increase in density and adiabatic bulk modulus as was calculated and shown in our earlier study [16]. Accordingly, the phase transition in olivine cannot fully explain the seismic boundary at 520 km. Moreover, the olivine–wadsleyite transition at the depth 410 km at 14–15 GPa calculated in [16] corresponded to a temperature of ~1700 K [16], but according to the estimations in [85], the temperature in this depth is 1790 to 1830 K. The calculated phase transition of (larnite + CaSi₂O₅)–Ca-perovskite occurs at a depth of 410 km at temperatures 1700–1800 K (Figure 6). Although the phase transitions in the CaSiO₃ system cannot be directly responsible for the seismic boundaries in the upper mantle, they have an effect on the location of discontinuities and point to the role of calcium-containing minerals in the upper mantle.

It is well known that magnesium is the most abundant element in mantle minerals, but the ratios of main elements Mg, Fe, Ca, and Al remain an open question. The calculated density of MgSiO₃-perovskite from the mantle geotherm [83] is too low with respect to the PREM model. However, the calculated density of CaSiO₃-perovskite is in good agreement with the density by the PREM model (Figure 8a). The calcium content in the perovskite composition will increase the density of mineral, as well as the presence of iron. Such a composition will obviously be close to the real composition of the mantle material. The obtained thermodynamic data in the present study can be used in the calculation of silicate mixtures of different compositions, as was done for example in [86]. Thus, the possibilities of thermodynamic modeling for future calculations and investigation of deep mineralogy in the area that is not available for direct experimentation are opened.

5. Conclusions

We have constructed the equations of state of wollastonite, pseudowollastonite, breyite, larnite (Ca₂SiO₄), titanite-structured CaSi₂O₅ and CaSiO₃-perovskite using a thermodynamic model based on the Helmholtz free energy. The proposed equations of state are developed based on optimization of different experimental measurements at ambient conditions and to a high temperatures and pressures. The full set of thermodynamic properties (volume, thermal expansion, entropy, isobaric and isochoric heat capacity, bulk moduli, Gibbs energy, etc.) of studied Ca-silicates is calculated at given pressures and temperatures and is presented in the form of Tables to the present article. The calculated properties are compared with reference and experimental data. It is shown that the proposed equations of state reliably describe properties of studied Ca-silicates in a wide range of pressures and temperatures.

The phase diagram of the CaSiO₃ system is constructed at pressures up to 20 GPa and temperatures up to 2000 K. The calculated phase diagram are compared with experimental data for different phases and clarifies the phase boundaries in the CaSiO₃ system under upper mantle conditions. The phase transition of majorite garnet, which contains CaSiO₃ phase in its composition, to Ca-perovskite is associated with seismic boundary at the depth 520 km; therefore, it points to the role of calcium-containing minerals in the upper mantle region. The calculated density of CaSiO₃-perovskite is in good agreement with the density according to the PREM model at the lower-mantle region. The calculated thermodynamic data for Ca-perovskite open the possibilities of thermodynamic modeling at high pressures and temperatures.